The behavior of depth functions of cover ideals of unimodular hypergraphs

Abstract

We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing depth function property. Furthermore, we show that the index of depth stability of these ideals is bounded by the number of variables. Introduction Let R=k[x1, ..., xn] be a polynomial ring over a given field k, and let I be a homogeneous ideal in R. It is known by Brodmann [3] that depth(R/I) takes a constant value for large s. Moreover, lim s→∞ depthR/I dimR− (I), where (I) is the analytic spread of I. The index of depth stability of I is defined by dstab(I) :=min { s0 1 |depthS/I =depthS/I0 for all s s0 } . Two natural questions arise from Brodmann’s theorem: (1) What is the nature of the function s →depthR/Is for s dstab(I)? (2) What is a reasonable bound for dstab(I)? On the nature of the function s →depthR/Is for s 1, which is called the depth function of I, Herzog and Hibi [10] conjectured that the depth function of ideals can be any convergent nonnegative integer valued function. The answer is affirmative for bounded increasing functions (see [10]) and non-increasing functions (see [8]). The behavior of depth functions, even for monomial ideals, is complicated (see e.g. [1]). Squarefree monomial ideals behave considerably better than monomial